
We may also investigate the higher-order singular triplets
\((\sigma _j, \, \phi _j, \, \psi _j)\)  that satisfy equivalent conditions
with the additional requirement of orthogonality of \(\lbrace \phi _j\rbrace \)  and \(\lbrace \psi _j\rbrace \) .
Equivalently, we wish to determine the Hilbert–Schmidt decomposition of \(R_\omega \) 
[1]}:
\(R_\omega = \sum _{j = 1}^\infty \sigma _j \psi _j(\xi ) \left<\phi _j,\, \cdot \right>,\) 
